What is the integral of (x-3)/(x^3-2x^2+3x) ?

The integral of (x-3)/(x^3-2x^2+3x) is

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integral of (x-3)/(x^3-2x^2+3x)

\int \frac{x - 3}{x ^{3} - 2 x ^{2} + 3 x} d x

\frac{1}{2} arctan (\frac{1}{2} (x - 1)) - ln ∣ x ∣ + \frac{1}{2} ln x^{2} - 2 x + 3 + \frac{1}{2} arctan (\frac{1}{2} (x - 1)) - 2 arctan (\frac{1}{2} (x - 1)) + C

Solution steps

\int \frac{x - 3}{x ^{3} - 2 x ^{2} + 3 x} d x

Expand

\frac{x - 3}{x ^{3} - 2 x ^{2} + 3 x} : \frac{x}{x ^{3} - 2 x ^{2} + 3 x} - \frac{3}{x ^{3} - 2 x ^{2} + 3 x}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{x ^{3} - 2 x ^{2} + 3 x} d x - \int \frac{3}{x ^{3} - 2 x ^{2} + 3 x} d x

\int \frac{x}{x ^{3} - 2 x ^{2} + 3 x} d x = \frac{1}{2} arctan (\frac{x - 1}{2})

\int \frac{3}{x ^{3} - 2 x ^{2} + 3 x} d x = ln ∣ x ∣ - \frac{1}{2} ln x^{2} - 2 x + 3 - \frac{1}{2} arctan (\frac{1}{2} (x - 1)) + 2 arctan (\frac{1}{2} (x - 1))

= \frac{1}{2} arctan (\frac{x - 1}{2}) - (ln ∣ x ∣ - \frac{1}{2} ln x^{2} - 2 x + 3 - \frac{1}{2} arctan (\frac{1}{2} (x - 1)) + 2 arctan (\frac{1}{2} (x - 1)))

Simplify

\frac{1}{2} arctan (\frac{x - 1}{2}) - (ln ∣ x ∣ - \frac{1}{2} ln x^{2} - 2 x + 3 - \frac{1}{2} arctan (\frac{1}{2} (x - 1)) + 2 arctan (\frac{1}{2} (x - 1))) : \frac{1}{2} arctan (\frac{1}{2} (x - 1)) - ln ∣ x ∣ + \frac{1}{2} ln x^{2} - 2 x + 3 + \frac{1}{2} arctan (\frac{1}{2} (x - 1)) - 2 arctan (\frac{1}{2} (x - 1))

= \frac{1}{2} arctan (\frac{1}{2} (x - 1)) - ln ∣ x ∣ + \frac{1}{2} ln x^{2} - 2 x + 3 + \frac{1}{2} arctan (\frac{1}{2} (x - 1)) - 2 arctan (\frac{1}{2} (x - 1))

Add a constant to the solution

= \frac{1}{2} arctan (\frac{1}{2} (x - 1)) - ln ∣ x ∣ + \frac{1}{2} ln x^{2} - 2 x + 3 + \frac{1}{2} arctan (\frac{1}{2} (x - 1)) - 2 arctan (\frac{1}{2} (x - 1)) + C

x \to 3 lim (\frac{1}{2 ^{x} - 8})

x \to 0 lim (\frac{x}{tan ( 5 x )})

\int \frac{sin ( 2 x )}{cos ^{2} ( 2 x )} d x

x \to 1 lim (x + 2)

f (x) = ln (3 x^{2} - 7 x + 4)

What is the integral of (x-3)/(x^3-2x^2+3x) ?
The integral of (x-3)/(x^3-2x^2+3x) is